The flop restriction complement is the finite ADE relative complement

Use the geometric slice input under the inherited assumption [F7]. Let $E$ be the geometric root space in [F1] and define the linear isomorphism $T:V\to E$ by $T({\alpha }_i)={\beta }_i$ for every label $i$. In the ADE Coxeter form, $2B({\alpha }_k,{\alpha }_i)$ equals $2$ on the diagonal, $-1$ on an edge because $m_{ik}=3$, and $0$ off the edges because $m_{ik}=2$. These are exactly the entries $a_{ik}$ in [F4]. Thus $T{r}_i({\alpha }_k)={\beta }_k-a_{ik}{\beta }_i=s_{i}T({\alpha }_k)$ for all $i,k$, so $T{r}_i=s_{i}T$ on the whole space. It follows by composition that $T$ sends the canonical simple-root orbits exactly to the geometric simple-root orbits, which exhaust the geometric root system by the simple-reflection generation assertion in [F5]. Hence $T(\mathrm{\Phi })$ is the full geometric root set. This basis calculation works separately on every component and requires no common scalar relating different component normalizations.

On dual spaces the map $f\mapsto g=f\circ T^{-1}$ satisfies $g({\beta }_i)=f({\alpha }_i)$ for every $i$. It therefore preserves all labelled simple-root dual coordinates, sends ${\mathrm{\Theta }}_J$ to $L$, and sends $H_{\alpha }$ to the hyperplane $g(T\alpha )=0$. By step 1.1 these are exactly the geometric root hyperplanes in [F1]. Containment of the respective coordinate subspace is preserved by this bijection, so the distinct proper restrictions are precisely the two relative hyperplane collections. Projecting $L$ to its coordinates indexed by $I$ is a linear isomorphism $L\to {\mathbb{R}}^n$, with inverse extension by zero on $J$; compose it with the dual map to obtain $q:{\mathrm{\Theta }}_J\to {\mathbb{R}}^n$. Its coordinate formulas and inverse are continuous, so it is a homeomorphism identifying the hyperplanes with ${\mathcal{H}}_{\mathrm{\Lambda }}$.

Since $W$ is finite, [F2] gives $\mathrm{Cone}=\mathrm{\Theta }$. Thus $\mathrm{Cone}(J)={\mathrm{\Theta }}_J$ and its interior in ${\mathrm{\Theta }}_J$ is all of ${\mathrm{\Theta }}_J$. Together with step 2.1 this proves equality of the real domains as well as the arrangement identification.

The map $(x,y)\mapsto q(x)+iq(y)$ is a real-linear homeomorphism ${\mathrm{\Theta }}_J\times {\mathrm{\Theta }}_J\to {\mathbb{C}}^n$, with inverse obtained from real and imaginary parts and $q^{-1}$. For each relative hyperplane $H$, its image on $H\times H$ is $q(H{)}_{\mathbb{C}}$. It therefore sends the deleted union in [F3] exactly to the displayed union and restricts to the asserted homeomorphism of complements. Here $n=\mathrm{\mid }I\mathrm{\mid }$ because precisely those coordinates remain free in $L$.

By step 2.1 the real complement homeomorphism gives a bijection of connected components, hence of chambers in [F6]. The linear isomorphism $q$ sends each ambient chamber closure to its image closure, sends its intersection with $H$ to the image closure intersected with $q(H)$, and preserves affine dimension. It thus sends walls bijectively to walls and preserves sharing a wall in both directions. Applying this chamber bijection term by term sends every chamber path in the Statement to one of the same length, with the inverse induced by $q^{-1}$. For each fixed pair of endpoints the sets of available path lengths are identical, so any minimum is the same.